Time-fractional cocaine–heroin dynamics with spatially heterogeneous diffusion: analysis and fractional Parker–Sochacki approximation

Résumé

Fractional reaction–diffusion models provide a powerful framework for describing dynamical systems in which memory effects and spatial interactions are significant. In this study, we develop and analyze a spatially heterogeneous time-fractional reaction–diffusion model for the coupled dynamics of cocaine and heroin abuse. Memory effects associated with addiction persistence, delayed behavioral responses, and relapse are incorporated via Caputo time-fractional derivatives, whereas spatial heterogeneity is represented by space-dependent diffusion coefficients, including both smoothly varying diffusion profiles and multi-patch configurations that capture regional disparities. To efficiently approximate the resulting nonlinear fractional system, we propose a fractional extension of the Parker–Sochacki method for time-fractional reaction–diffusion equations with heterogeneous diffusion. The proposed scheme provides an explicit and computationally efficient numerical algorithm that significantly reduces the substantial memory requirements typically associated with fractional-order models. The mathematical properties of the model are investigated rigorously. The existence, uniqueness, positivity, and boundedness of solutions are established using the theory of sectorial operators, and the equilibrium states are characterized. Furthermore, global stability conditions are derived using appropriate Lyapunov functionals. Numerical simulations validate the proposed approach and demonstrate the combined effects of memory and spatial heterogeneity on cocaine–heroin dynamics. In particular, lower fractional orders are shown to promote prolonged persistence of substance abuse, while heterogeneous diffusion generates pronounced nonuniform spatial distributions. The proposed framework integrates methodological advances with epidemiological insight, offering a robust tool for studying substance-use dynamics in spatially heterogeneous environments.

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